Abstract
The visibility representation (VR for short) is a classical representation of plane graphs. The VR has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. It is known that there exists a plane graph G with n vertices where any VR of G requires a size at least \(\lfloor \frac{2n}{3} \rfloor \times (\lfloor \frac{4n}{3} \rfloor -3)\). For upper bounds, it is known that every plane graph has a VR with size at most \(\lfloor \frac{2}{3}n \rfloor \times (2n-5)\), and a VR with size at most \((n-1) \times \lfloor \frac{4}{3}n \rfloor\).
It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely of size c h n ×c w n with c h < 1 and c w < 2). In this paper, we provide the first VR construction for a non-trivial graph class that simultaneously bounds both the height and the width. We prove that every 4-connected plane graph has a VR with height \(\leq \frac{3n}{4}+2\lceil\sqrt{n}\rceil +4\) and width \(\leq \lceil \frac{3n}{2}\rceil\). Our VR algorithm is based on an st-orientation of 4-connected plane graphs with special properties. Since the st-orientation is a very useful concept in other applications, this result may be of independent interests.
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He, X., Wang, JJ., Zhang, H. (2010). Compact Visibility Representation of 4-Connected Plane Graphs. In: Wu, W., Daescu, O. (eds) Combinatorial Optimization and Applications. COCOA 2010. Lecture Notes in Computer Science, vol 6508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17458-2_28
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DOI: https://doi.org/10.1007/978-3-642-17458-2_28
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